n^2+6n-77=0

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Solution for n^2+6n-77=0 equation:


Simplifying
n2 + 6n + -77 = 0

Reorder the terms:
-77 + 6n + n2 = 0

Solving
-77 + 6n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '77' to each side of the equation.
-77 + 6n + 77 + n2 = 0 + 77

Reorder the terms:
-77 + 77 + 6n + n2 = 0 + 77

Combine like terms: -77 + 77 = 0
0 + 6n + n2 = 0 + 77
6n + n2 = 0 + 77

Combine like terms: 0 + 77 = 77
6n + n2 = 77

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = 77 + 9

Reorder the terms:
9 + 6n + n2 = 77 + 9

Combine like terms: 77 + 9 = 86
9 + 6n + n2 = 86

Factor a perfect square on the left side:
(n + 3)(n + 3) = 86

Calculate the square root of the right side: 9.273618495

Break this problem into two subproblems by setting 
(n + 3) equal to 9.273618495 and -9.273618495.

Subproblem 1

n + 3 = 9.273618495 Simplifying n + 3 = 9.273618495 Reorder the terms: 3 + n = 9.273618495 Solving 3 + n = 9.273618495 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 9.273618495 + -3 Combine like terms: 3 + -3 = 0 0 + n = 9.273618495 + -3 n = 9.273618495 + -3 Combine like terms: 9.273618495 + -3 = 6.273618495 n = 6.273618495 Simplifying n = 6.273618495

Subproblem 2

n + 3 = -9.273618495 Simplifying n + 3 = -9.273618495 Reorder the terms: 3 + n = -9.273618495 Solving 3 + n = -9.273618495 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -9.273618495 + -3 Combine like terms: 3 + -3 = 0 0 + n = -9.273618495 + -3 n = -9.273618495 + -3 Combine like terms: -9.273618495 + -3 = -12.273618495 n = -12.273618495 Simplifying n = -12.273618495

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.273618495, -12.273618495}

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